Role of Feedback in Some Models for Tumour Growth
Philip K. Maini
Centre for Mathematical Biology Mathematical Institute;
Oxford Centre for Integrative Systems Biology, Biochemistry
Outline
• A very simple ODE model for cell
population dynamics in intestinal crypts
• An individual-based model for the above
• A multiscale model for vascularised tumour growth
Colorectal Cancer
• Second leading cause of cancer deaths in the US
Cell population dynamics in the colonic crypt
• Johnston, Edwards, Bodmer, PKM, Chapman, PNAS, 104, 4008-4013 (2007)
• Crypt can be considered as 3-compartments:
Stem cells,
Semi-Differentiated (transit-amplifying) Cells, Fully Differentiated Cells
A previous model
• Tomlinson and Bodmer, PNAS, 92, 11130- 11134 (1995) – proposed cell cycle
synchrony and no feedback
• D’Onofrio, Tomlinson, JTB, 244, 367-374 (2006) – feedback, but still cell synchrony
• Both ignore the compounding effect of TA cells cycling faster than stem cells
Continuous Model
• Interested on time scales greater than the cell cycle time – continuous cell division
Steady States
• For both these models, non-trivial steady states (corresponding to homeostatis) only occur if the parameters take specific
values --- STRUCTURALLY UNSTABLE
Need Feedback
• Wodarz 2007 (mutations) – feedback in stem cell compartment
• Boman et al 2001 – no feedback in stem cells so they tend to zero
• Komarova – series on papers on mutations
Model 1 – Linear Feedback
• Assume that when the population of stem or TA cells increase, the per capita rate at which they differentiate increases in
proportion
Steady States
• Stem cells exhibit logistic growth – nice bounded stable steady state solution
• For a very large region in parameter space this model predicts homeostatis
• Only a genetic hit which removes the feedback in the model will lead to
unbounded growth
Model 2 – Saturating Feedback
• Assume maximum per-capita rate of differentiation
Steady States
• Nice homeostatic steady state as long as net linear growth rate of stem cells lies in a certain region in parameter space. If a
mutation moves us out of this region then the population grows unbounded, even in the presence of feedback
• Therefore the proportion of cancer-driving cells may vary greatly from tumour to
tumour (Johnston, Edwards,Chapman, PKM, Bodmer, JTB online 2010)
• Nature Reports Stem Cells:
• Cancer stem cells, becoming common, M.
Baker, 3/12/08
• Lander, Nie and Wan: olfactory endothelium
Some bad things
• Continuum approximation – individual- based model approach in IB
• Handling of mutations – properly including mutant population
Conclusions
• Developed a robust model for cell populations in the crypt
• Shown that the key parameters are the net per-capita growth rates of the stem cells and TA cells. So, the failure of programmed cell death or differentiation could lead to tumour growth, as well as increased proliferation rate
• Saturating feedback could explain the existence of benign tumours before carcinogenesis takes over – early mutations could keep
parameters below their critical values, later stage mutations could push them above their critical values.
• Evidence suggests that nearly all colorectal cancers go through benign stages, but not all develop into carcinomas
An integrative computational model for intestinal tissue renewal
• Van Leeuwen, Mirams, Walter, Fletcher, Murray, Osbourne, Varma, Young,
Cooper, Pitt-Francis, Momtahan,
Pathmanathan, Whiteley, Chapman,
Gavaghan, Jensen, King, PKM, Waters, Byrne (Cell Proliferation, 2009)
• Chaste – Cancer, Heart And Soft Tissue Environment
• Modular
• Follow Meinke et al and use a cell-centred approach – basically consider cells as
point masses connected by springs and use Delauny triangularisation and Voronoi tesselation to determine nearest
neighbours and for visualisation.
The effects of different individual cell-based approaches
• (to appear in Phil Trans R Soc A)
Cancer Growth
Tissue Level Signalling: (Tumour Angiogenesis Factors) Oxygen etc
Cells:
Intracellular: Cell cycle, Molecular elements
Partial Differential Equations Automaton Elements
Ordinary differential equations
• Tomas Alarcon
• Markus Owen
• Helen Byrne
• James Murphy
• Russel Bettridge
Vascular Adaptation
• Series of papers by Secomb and Pries modelling vessels in the rat mesentry – they conclude:
R(t) = radius at time t:
R(t+dt) = R(t) + R dt S
S = M + Me – s + C
M = mechanical stimulus (wall shear stress) Me = metabolic demand
s = shrinkage
C= conducted stimuli: short-range (chemical release under hypoxic stress?)
long-range (mediated through membrane potential?)
• By varying the strengths of the different
adaptation mechanisms we can hypothesise
how defects in vasculature lead to different types of tumours: Conclude that losing the long range stimuli looks a reasonable assumption
• Tim Secomb has shown this more convincingly recently (PLoS Comp Biol 2009)
Potential uses of the model
• Chemotherapy
• Impact of cell crowding and active movement
• Vessel normalisation
Angiogenesis
• Recently, we have added in angiogenesis (Owen, Alarcon, PKM and Byrne, J.Math.
Biol, 09)
• Movie – both2_mov
So what?
• Mark Lloyd (Moffitt) is now doing
experiments on cancer angiogenesis in mice to allow us to derive data for model testing.
Conclusions and Criticisms
• Simple multiscale model – gain some insight into why combination therapies might work
• Heterogeneities in environment play a key role
• No matrix included! – Anderson has shown adhesivity could be important
• Cellular automaton model – what about using Potts model, cell centred, cell vertex models? – DOES IT MAKE A DIFFERENCE (Murray et al, 2009; Byrne et al, 2010)
• There are many other models and I have not referred to any of them! (Jiang, Bauer, Chaplain, Anderson, Lowengrub, Drasdo, Meyer-Hermann, Rieger, Cristini, Enderling, Meinke, Loeffler, TO NAME BUT A FEW)
Acknowledgements
• Integrative Biology Project (EPSRC)
• NCI (NIH)
• RosBnet